Embedded newform 403.2.bl.b.16.19

• Label: 403.2.bl.b.16.19
• Level: $403$
• Weight: $2$
• Character: 403.16
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bl (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			16.19
Character	\(\chi\)	\(=\)	403.16
Dual form			403.2.bl.b.126.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.192109 + 0.0855325i) q^{2} +(0.298908 - 2.84392i) q^{3} +(-1.30867 - 1.45343i) q^{4} -1.10354 q^{5} +(0.300670 - 0.520777i) q^{6} +(0.556681 + 0.618257i) q^{7} +(-0.257059 - 0.791146i) q^{8} +(-5.06408 - 1.07640i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q + q^{2} + 39 q^{4} - 8 q^{5} - 2 q^{7} + 12 q^{8} + 29 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{1}{5}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.192109	+	0.0855325i	0.135842	+	0.0604806i	0.473532	−	0.880777i	\(-0.342979\pi\)
							−0.337690	+	0.941257i	\(0.609646\pi\)
\(3\)	0.298908	−	2.84392i	0.172574	−	1.64194i	−0.475037	−	0.879966i	\(-0.657565\pi\)
							0.647611	−	0.761971i	\(-0.275768\pi\)
\(5\)	−1.10354			−0.493518			−0.246759	−	0.969077i	\(-0.579366\pi\)
							−0.246759	+	0.969077i	\(0.579366\pi\)
\(7\)	0.556681	+	0.618257i	0.210406	+	0.233679i	0.839105	−	0.543969i	\(-0.183079\pi\)
							−0.628700	+	0.777648i	\(0.716412\pi\)
\(11\)	−4.31024	+	0.916169i	−1.29959	+	0.276235i	−0.805198	−	0.593006i	\(-0.797941\pi\)
							−0.494387	+	0.869242i	\(0.664608\pi\)
\(13\)	0.218340	−	3.59893i	0.0605567	−	0.998165i
							\(17\)	7.87533	+	1.67395i	1.91005	+	0.405993i	0.999964	−	0.00843182i	\(-0.00268396\pi\)
0.910083	+	0.414425i	\(0.136017\pi\)
\(19\)	0.249921	+	2.37784i	0.0573358	+	0.545514i	0.985056	+	0.172237i	\(0.0550994\pi\)
							−0.927720	+	0.373277i	\(0.878234\pi\)
\(23\)	3.02162	−	3.35585i	0.630051	−	0.699743i	−0.340607	−	0.940206i	\(-0.610633\pi\)
							0.970658	+	0.240463i	\(0.0772992\pi\)
\(29\)	−4.50948	−	2.00775i	−0.837389	−	0.372830i	−0.0571917	−	0.998363i	\(-0.518215\pi\)
							−0.780197	+	0.625534i	\(0.784881\pi\)
\(31\)	−5.03931	−	2.36755i	−0.905088	−	0.425225i
							\(37\)	−2.54264	−	4.40398i	−0.418007	−	0.724009i	0.577732	−	0.816227i	\(-0.303938\pi\)
−0.995739	+	0.0922174i	\(0.970605\pi\)
\(41\)	−5.21095	−	2.32006i	−0.813813	−	0.362333i	−0.0427453	−	0.999086i	\(-0.513610\pi\)
							−0.771068	+	0.636753i	\(0.780277\pi\)
\(43\)	−1.14508	−	10.8947i	−0.174622	−	1.66142i	−0.634115	−	0.773239i	\(-0.718635\pi\)
							0.459493	−	0.888182i	\(-0.348031\pi\)
\(47\)	8.60198	+	6.24970i	1.25473	+	0.911613i	0.998486	−	0.0549998i	\(-0.0175158\pi\)
							0.256241	+	0.966613i	\(0.417516\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bl.b.16.19	✓	280
13.9	even	3	inner	403.2.bl.b.295.17	yes	280
31.2	even	5	inner	403.2.bl.b.250.17	yes	280
403.126	even	15	inner	403.2.bl.b.126.19	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bl.b.16.19	✓	280	1.1	even	1	trivial
403.2.bl.b.126.19	yes	280	403.126	even	15	inner
403.2.bl.b.250.17	yes	280	31.2	even	5	inner
403.2.bl.b.295.17	yes	280	13.9	even	3	inner